Complex reactions polynomial kinetics

The non-linear theory of steady-steady (quasi-steady-state/pseudo-steady-state) kinetics of complex catalytic reactions is developed. It is illustrated in detail by the example of the single-route reversible catalytic reaction. The theoretical framework is based on the concept of the kinetic polynomial which has been proposed by authors in 1980-1990s and recent results of the algebraic theory, i.e. an approach of hypergeometric functions introduced by Gel fand, Kapranov and Zelevinsky (1994) and more developed recently by Sturnfels (2000) and Passare and Tsikh (2004). The concept of ensemble of equilibrium subsystems introduced in our earlier papers (see in detail Lazman and Yablonskii, 1991) was used as a physico-chemical and mathematical tool, which generalizes the well-known concept of equilibrium step . In each equilibrium subsystem, (n—1) steps are considered to be under equilibrium conditions and one step is limiting n is a number of steps of the complex reaction). It was shown that all solutions of these equilibrium subsystems define coefficients of the kinetic polynomial. 

Figure 1 Dependence of overall reaction rate on the parameter 2 (LH mechanism). Branches Rl, R2, R3 and R4 represent the roots of kinetic polynomial. Solid line indicates feasible steady states. Branches Re(Rl), Re(R2) and Re(R3) correspond to the real parts of conjugated complex roots of kinetic polynomial. Parameter values fi = 1.4, — 0.1, t2 = 0.1, fj = 15 and rj = 2.

Applying "kinetic polynomial" approach we found the analytical representation for the "thermodynamic branch" of the overall reaction rate of the complex reaction with no traditional assumptions on the rate limiting and "fast" equilibrium of steps. 

Calculation of the coefficients dt for a given matrix is a very laborious process. We will give a method to calculate these coefficients proceeding directly from the complex reaction graph. Like a steady-state kinetic equation, a characteristic polynomial will be represented in the general (struc-turalized) form ... 

Finally, it appears that the kinetic models of complex reactions contain two types of components independent of and dependent on the complex mechanism structure [4—7]. Hence the thermodynamic correctness of these models is ensured. The analysis of simple classes indicates that an unusual analog arises for the equation of state relating the observed characteristics of the open chemical system, i.e. a kinetic polynomial [7]. This polynomial distinctly shows how a complex kinetic relationship is assembled from simple reaction equations. 

For many catalytic reactions with nonlinear steps, derivation of kinetic equations can be challenging. In order to avoid such difficulties, Lazman and Yablonsky applied constructive algebraic geometry to nonlinear kinetics, expressing the reaction rate of a complex reaction as an implicit function of concentrations and temperature. This concept of kinetic polynomial [6] has found important applications including parameter estimation, analysis of kinetic model identifiability and finding all steady-states of kinetic models. The Lazman-Yablonsky four-term rate equation for the polynomial kinetics is ... 

Kinetic polynomials, especially in the case of complex reaction mechanisms, are quite complicated. Just to illustrate this point, let us consider the Langmuir Hinshelwood mechanism ... 

Equation (3) is linear with respect to the reaction rate variable, R. In the further analysis of more complex, non-linear, mechanisms and corresponding kinetic models, we will present the polynomial as an equation, which generalizes Equation (3), and term it as the kinetic polynomial. We will demonstrate that the overall reaction rate, in the general non-linear case, cannot generally be presented as a difference between two terms representing the forward and reverse reaction rates. This presentation is valid only at the special conditions that will be described. 

A non-linear theory of steady-state kinetics of complex catalytic reactions is developed. A system of steady-state (or pseudo-steady-state) equations can always be reduced to a so called kinetic polynomial. This polynomial is a function of the steady-state reaction rate and the process parameters (concentrations of the reactants, temperature). 

Building on similar ideas but starting from a more detailed reaction scheme, Tyson (1991) proposed a model for the mitotic oscillator based on the formation of a complex between cycUn and cdc2 kinase, followed by the activation of this complex. Essential to the oscillatory mechanism is the assumption that the active complex, i.e. MPF, promotes its own activation in a nonlinear memner. The kinetic equations, of a polynomial form, reduce under some simplifying assumptions to the equations of the two-variable Brusselator model. Inactivation of MPF is not... 

Now the more complex defect equilibria are treated with the aid of numerical methods. Spears and Tuller show how this approach can be used, not only to extract principal defect thermodynamic and kinetic data, but also to assist in the design and optimization of materials. Using all possible defect reactions and defect equilibria, combined with Equation (5.86), one arrives at an eight-order polynomial in, for instance, [e ], which can only be solved numerically. In a simplified approach for the nominally pure material, expressions for [e ] and [h ] are obtained, which are concordant to Equations (5.51) and (5.52). The simple Ktbger- nk diagram for such a defect situation has already been presented by Kofstad. ... 

Eq. (4.179) contains (1) the kinetic apparent coefficient k+ (2) the potential term, or driving force related to the thermodynamics of the net reaction (3) the term of resistance, ie, the denominator, which reflects the complexity of reaction, both its multi-step character and its non-hnearity finally, (4) the non-linear term (N k,c)), a polynomial in concentrations and kinetic parameters, which is caused exclusively by nonlinear steps. In the case of a linear mechanism, this term vanishes. In classical kinetics of heterogeneous catalysis (LHHW equations), such term is absent. 

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